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Question Number 174276 by cortano1 last updated on 28/Jul/22

Answered by blackmamba last updated on 28/Jul/22

$$\underset{x\to 0}{\lim }\frac{1-\sqrt[2022]{\cos \left(2021x\right)}}{x^2}=$$$$\underset{x\to 0}{\lim }\frac{1-\sqrt[2022]{1-2\sin {}^2\left(\frac{2021x}{2}\right)}}{x^2}$$$$=\underset{x\to 0}{\lim }\frac{1-(1-\frac{2\sin {}^2\left(\frac{2021x}{2}\right)}{2022})}{x^2}$$$$=\frac{1}{1011}\times \underset{x\to 0}{\lim }{\left[\frac{\sin \left(\frac{2021x}{2}\right)}{x}\right]}^2$$$$=\frac{1}{1011}\times \frac{{\left(2021\right)}^2}{4}=\frac{{2021}^2}{4044}$$

Answered by Mathspace last updated on 28/Jul/22

$$f\left(x\right)=\frac{1-{\left(cos\left(2021x\right)\right)}^{\frac{1}{2022}}}{x^2}$$$$wehavecosx\left(\alpha x\right)\sim 1-\frac{{\alpha }^2{x}^2}{2}$$$$\Rightarrow {\left(cos\left(\alpha x\right)\right)}^m\sim {(1-\frac{{\alpha }^2{x}^2}{2})}^m$$$$\sim 1-\frac{m{\alpha }^2}{2}\Rightarrow f\left(x\right)\sim \frac{1-(1-1+\frac{1}{2022}{\left(2021\right)}^2\frac{x^2}{2})}{x^2}$$$$\Rightarrow f\left(x\right)\sim \frac{{\left(2021\right)}^2}{2\times 2022}(x\to 0)\Rightarrow$$$${lim}_{x\to 0}f\left(x\right)=\frac{{\left(2021\right)}^2}{4044}$$

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